Do unschooled people use cases correctly, e.g. in Germany and in Russia?

Morphological complexity as such as is not related to the level of schooling. Some of the most morphologically complex languages are spoken by people without any education. So, all Russian and German speakers (including those with no formal schooling) use the morphological cases in their respective languages. So did speakers of Vulgar Latin which was really just a vernacular counterpart of Standard Latin.

What you're referring to is the difference between standard and non-standard dialects which will differ in the use of cases, morphology, etc. This may be identified by speakers of the standard as a 'lack of cases'.

In short, what is related to schooling is the ability to speak the standard dialect not the ability to use complex morphology as such.